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Topic: Simple(?) theoretical set theory questions
Replies: 6   Last Post: Jun 16, 1996 3:34 AM

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Dan Wevrick

Posts: 55
Registered: 12/6/04
Simple(?) theoretical set theory questions
Posted: Jun 13, 1996 10:20 AM
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I'm involved in a bit of discussion over in alt.algebra.help and thought
I'd try to clarify some things we've been discussing.

Q1: What is the commonly accepted definition of two sets having the same
cardinality?
Is it "the existence of a bijection between them" ?

Q2: Do you need the Axiom of Choice to prove that every infinite
set has a countable subset?
My feeling is no, since the following should work: Let A be the
set. A is non-empty so choose a_1 in A (no axiom of choice needed,
a_1 exists since A is non-empty). Now A\{a_1} is infinite (otherwise
A would be finite) and hence a_2 exists in A\{a_1}. Repeat (ie.induction)
to get a_n for all natural N, ie. a countable subset.

Q3: Do you need the Axiom of Choice to show the following: For any two
sets A and B, either there exists an injection from A to B or an injection
from B to A

Thanks in advance,
Dan







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